Optimal. Leaf size=87 \[ -\frac {(a C+b B) \cot ^2(c+d x)}{2 d}+\frac {(a B-b C) \cot (c+d x)}{d}-\frac {(a C+b B) \log (\sin (c+d x))}{d}+x (a B-b C)-\frac {a B \cot ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.19, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {3632, 3591, 3529, 3531, 3475} \[ -\frac {(a C+b B) \cot ^2(c+d x)}{2 d}+\frac {(a B-b C) \cot (c+d x)}{d}-\frac {(a C+b B) \log (\sin (c+d x))}{d}+x (a B-b C)-\frac {a B \cot ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rule 3591
Rule 3632
Rubi steps
\begin {align*} \int \cot ^5(c+d x) (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\int \cot ^4(c+d x) (a+b \tan (c+d x)) (B+C \tan (c+d x)) \, dx\\ &=-\frac {a B \cot ^3(c+d x)}{3 d}+\int \cot ^3(c+d x) (b B+a C-(a B-b C) \tan (c+d x)) \, dx\\ &=-\frac {(b B+a C) \cot ^2(c+d x)}{2 d}-\frac {a B \cot ^3(c+d x)}{3 d}+\int \cot ^2(c+d x) (-a B+b C-(b B+a C) \tan (c+d x)) \, dx\\ &=\frac {(a B-b C) \cot (c+d x)}{d}-\frac {(b B+a C) \cot ^2(c+d x)}{2 d}-\frac {a B \cot ^3(c+d x)}{3 d}+\int \cot (c+d x) (-b B-a C+(a B-b C) \tan (c+d x)) \, dx\\ &=(a B-b C) x+\frac {(a B-b C) \cot (c+d x)}{d}-\frac {(b B+a C) \cot ^2(c+d x)}{2 d}-\frac {a B \cot ^3(c+d x)}{3 d}+(-b B-a C) \int \cot (c+d x) \, dx\\ &=(a B-b C) x+\frac {(a B-b C) \cot (c+d x)}{d}-\frac {(b B+a C) \cot ^2(c+d x)}{2 d}-\frac {a B \cot ^3(c+d x)}{3 d}-\frac {(b B+a C) \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 1.03, size = 101, normalized size = 1.16 \[ -\frac {3 (a C+b B) \left (\cot ^2(c+d x)+2 (\log (\tan (c+d x))+\log (\cos (c+d x)))\right )+2 a B \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )+6 b C \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 121, normalized size = 1.39 \[ -\frac {3 \, {\left (C a + B b\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} - 3 \, {\left (2 \, {\left (B a - C b\right )} d x - C a - B b\right )} \tan \left (d x + c\right )^{3} - 6 \, {\left (B a - C b\right )} \tan \left (d x + c\right )^{2} + 2 \, B a + 3 \, {\left (C a + B b\right )} \tan \left (d x + c\right )}{6 \, d \tan \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 7.62, size = 237, normalized size = 2.72 \[ \frac {B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, {\left (B a - C b\right )} {\left (d x + c\right )} + 24 \, {\left (C a + B b\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 24 \, {\left (C a + B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {44 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 44 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 124, normalized size = 1.43 \[ -\frac {a B \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a B \cot \left (d x +c \right )}{d}+a B x +\frac {B a c}{d}-\frac {a C \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a C \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {B b \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {B b \ln \left (\sin \left (d x +c \right )\right )}{d}-b C x -\frac {C \cot \left (d x +c \right ) b}{d}-\frac {C b c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 104, normalized size = 1.20 \[ \frac {6 \, {\left (B a - C b\right )} {\left (d x + c\right )} + 3 \, {\left (C a + B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, {\left (C a + B b\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {6 \, {\left (B a - C b\right )} \tan \left (d x + c\right )^{2} - 2 \, B a - 3 \, {\left (C a + B b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.89, size = 127, normalized size = 1.46 \[ -\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\left (C\,b-B\,a\right )\,{\mathrm {tan}\left (c+d\,x\right )}^2+\left (\frac {B\,b}{2}+\frac {C\,a}{2}\right )\,\mathrm {tan}\left (c+d\,x\right )+\frac {B\,a}{3}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,b+C\,a\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,\left (a+b\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,\left (b+a\,1{}\mathrm {i}\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.43, size = 180, normalized size = 2.07 \[ \begin {cases} \text {NaN} & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan {\relax (c )}\right ) \left (B \tan {\relax (c )} + C \tan ^{2}{\relax (c )}\right ) \cot ^{5}{\relax (c )} & \text {for}\: d = 0 \\B a x + \frac {B a}{d \tan {\left (c + d x \right )}} - \frac {B a}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {B b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {B b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {B b}{2 d \tan ^{2}{\left (c + d x \right )}} + \frac {C a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {C a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {C a}{2 d \tan ^{2}{\left (c + d x \right )}} - C b x - \frac {C b}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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